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My understanding was that relativistic physics can be expressed in any inertial coordinate system, but not arbitrary systems. That is, no experiment can determine if we are "still" or "moving" at a constant velocity; but we can determine if we are accelerating, or moving in a circle (which by definition involves constant acceleration perpendicular to the current velocity).

Thus, we can clearly state that the Earth is orbiting, and can't view it as relativisticly stationary.

But, to my shock, I recently came across this text http://books.google.com/books?id=lWEmNBaHCJMC&pg=PA211&dq=einstein+infeld+physics+ptolemy+copernicus&hl=en&ei=dWZ_TubbKqn20gH8hNjSDw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCwQ6AEwAA#v=onepage&q&f=false (which has Einstein as a coauthor) which, on page 212, seems to say that although special relativity requires an inertial coordinate system, general relativity does not! And that therefore we can state that the Earth is stationary and the Sun orbits it! I would reject this as pseudoscientific bunk, if not for the authors of the book.

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"on page 212, seems to say that although special relativity requires an inertial coordinate system, general relativity does not! And that therefore we can state that the Earth is stationary and the Sun orbits it!" - Your deduction (...therefore...) is incorrect. The fact that GR does not assume inertial coordinates does not mean what you write. –  mtrencseni Sep 26 '11 at 14:27
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My understanding was that relativistic physics can be expressed in any inertial coordinate system, but not arbitrary systems.

No, both SR and GR are capable of dealing with accelerated frames of reference. GR doesn't even have global frames of reference, and it's not valid to think of a coordinate system as being the same thing as a frame of reference. Ca. 1915, Einstein thought that GR should be interpreted as a generalization of SR to accelerated frames of reference. That was the wrong interpretation, and physicists today agree that the fundamental distinction between SR and GR is that SR can only deal with flat spacetime.

That is, no experiment can determine if we are "still" or "moving" at a constant velocity; but we can determine if we are accelerating

This is true, but not logically equivalent to the first statement. And what an accelerometer tells you is your acceleration relative to a local free-falling frame, which is what is known in GR as a local inertial frame (LIF). The notion of an LIF is different from the Newtonian notion of an inertial frame. In Newtonian physics, the earth's surface is an inertial frame. In GR, the earth's surface is not an LIF. (Newtonian frames are also global. GR doesn't have global frames.)

The Einstein field equations are written in a notation that is completely independent of the coordinates chosen. They have the same form for any choice of coordinates. This is unlike Newtonian physics, whose laws are simple when expressed in an inertial frame but more complex when expressed in a noninertial frame.

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